The value of `((sin theta + sin phi)/( cos theta + cos phi) + (cos theta - cos theta )/( sin theta - sin phi))` is

Options

A. 1
B. 2
C. `1/2`
D. 0

Answer

To solve the expression \(\frac{\sin \theta + \sin \phi}{\cos \theta + \cos \phi} + \frac{\cos \theta - \cos \phi}{\sin \theta - \sin \phi}\), we will follow these steps:

### Step 1: Write the expression
We start with the given expression:
\[ \frac{\sin \theta + \sin \phi}{\cos \theta + \cos \phi} + \frac{\cos \theta - \cos \phi}{\sin \theta - \sin \phi} \]

### Step 2: Find a common denominator
The common denominator for the two fractions is \((\cos \theta + \cos \phi)(\sin \theta - \sin \phi)\).

### Step 3: Rewrite the fractions
Now, we can rewrite the expression using the common denominator:
\[ \frac{(\sin \theta + \sin \phi)(\sin \theta - \sin \phi) + (\cos \theta - \cos \phi)(\cos \theta + \cos \phi)}{(\cos \theta + \cos \phi)(\sin \theta - \sin \phi)} \]

### Step 4: Expand the numerators
Now we expand the numerators:
1. For \((\sin \theta + \sin \phi)(\sin \theta - \sin \phi)\), we use the difference of squares:
\[ \sin^2 \theta - \sin^2 \phi \]
2. For \((\cos \theta - \cos \phi)(\cos \theta + \cos \phi)\), we also use the difference of squares:
\[ \cos^2 \theta - \cos^2 \phi \]

Combining these, we have:
\[ \sin^2 \theta - \sin^2 \phi + \cos^2 \theta - \cos^2 \phi \]

### Step 5: Use the Pythagorean identity
Using the identity \(\sin^2 x + \cos^2 x = 1\):
\[ (\sin^2 \theta + \cos^2 \theta) - (\sin^2 \phi + \cos^2 \phi) = 1 - 1 = 0 \]

### Step 6: Final expression
Now substituting back into our expression, we get:
\[ \frac{0}{(\cos \theta + \cos \phi)(\sin \theta - \sin \phi)} = 0 \]

### Conclusion
Thus, the value of the original expression is:
\[ \boxed{0} \]

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